Optimal. Leaf size=96 \[ \frac {a^3 A x^{1+m}}{1+m}+\frac {a^2 (3 A b+a B) x^{2+m}}{2+m}+\frac {3 a b (A b+a B) x^{3+m}}{3+m}+\frac {b^2 (A b+3 a B) x^{4+m}}{4+m}+\frac {b^3 B x^{5+m}}{5+m} \]
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Rubi [A]
time = 0.03, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77}
\begin {gather*} \frac {a^3 A x^{m+1}}{m+1}+\frac {a^2 x^{m+2} (a B+3 A b)}{m+2}+\frac {b^2 x^{m+4} (3 a B+A b)}{m+4}+\frac {3 a b x^{m+3} (a B+A b)}{m+3}+\frac {b^3 B x^{m+5}}{m+5} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int x^m (a+b x)^3 (A+B x) \, dx &=\int \left (a^3 A x^m+a^2 (3 A b+a B) x^{1+m}+3 a b (A b+a B) x^{2+m}+b^2 (A b+3 a B) x^{3+m}+b^3 B x^{4+m}\right ) \, dx\\ &=\frac {a^3 A x^{1+m}}{1+m}+\frac {a^2 (3 A b+a B) x^{2+m}}{2+m}+\frac {3 a b (A b+a B) x^{3+m}}{3+m}+\frac {b^2 (A b+3 a B) x^{4+m}}{4+m}+\frac {b^3 B x^{5+m}}{5+m}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 87, normalized size = 0.91 \begin {gather*} \frac {x^{1+m} \left (B (a+b x)^4+(-a B (1+m)+A b (5+m)) \left (\frac {a^3}{1+m}+\frac {3 a^2 b x}{2+m}+\frac {3 a b^2 x^2}{3+m}+\frac {b^3 x^3}{4+m}\right )\right )}{b (5+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 110, normalized size = 1.15
method | result | size |
norman | \(\frac {a^{2} \left (3 A b +B a \right ) x^{2} {\mathrm e}^{m \ln \left (x \right )}}{2+m}+\frac {a^{3} A x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {b^{2} \left (A b +3 B a \right ) x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}+\frac {b^{3} B \,x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}+\frac {3 a b \left (A b +B a \right ) x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}\) | \(110\) |
risch | \(\frac {x \left (B \,b^{3} m^{4} x^{4}+A \,b^{3} m^{4} x^{3}+3 B a \,b^{2} m^{4} x^{3}+10 B \,b^{3} m^{3} x^{4}+3 A a \,b^{2} m^{4} x^{2}+11 A \,b^{3} m^{3} x^{3}+3 B \,a^{2} b \,m^{4} x^{2}+33 B a \,b^{2} m^{3} x^{3}+35 B \,b^{3} m^{2} x^{4}+3 A \,a^{2} b \,m^{4} x +36 A a \,b^{2} m^{3} x^{2}+41 A \,b^{3} m^{2} x^{3}+B \,a^{3} m^{4} x +36 B \,a^{2} b \,m^{3} x^{2}+123 B a \,b^{2} m^{2} x^{3}+50 m \,x^{4} b^{3} B +A \,a^{3} m^{4}+39 A \,a^{2} b \,m^{3} x +147 A a \,b^{2} m^{2} x^{2}+61 A \,b^{3} x^{3} m +13 B \,a^{3} m^{3} x +147 B \,a^{2} b \,m^{2} x^{2}+183 B a \,b^{2} x^{3} m +24 b^{3} B \,x^{4}+14 A \,a^{3} m^{3}+177 A \,a^{2} b \,m^{2} x +234 a A \,b^{2} x^{2} m +30 A \,b^{3} x^{3}+59 B \,a^{3} m^{2} x +234 B \,a^{2} b \,x^{2} m +90 B a \,b^{2} x^{3}+71 A \,a^{3} m^{2}+321 a^{2} A b x m +120 a A \,b^{2} x^{2}+107 a^{3} B x m +120 B \,a^{2} b \,x^{2}+154 a^{3} A m +180 a^{2} A b x +60 a^{3} B x +120 a^{3} A \right ) x^{m}}{\left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(453\) |
gosper | \(\frac {x^{1+m} \left (B \,b^{3} m^{4} x^{4}+A \,b^{3} m^{4} x^{3}+3 B a \,b^{2} m^{4} x^{3}+10 B \,b^{3} m^{3} x^{4}+3 A a \,b^{2} m^{4} x^{2}+11 A \,b^{3} m^{3} x^{3}+3 B \,a^{2} b \,m^{4} x^{2}+33 B a \,b^{2} m^{3} x^{3}+35 B \,b^{3} m^{2} x^{4}+3 A \,a^{2} b \,m^{4} x +36 A a \,b^{2} m^{3} x^{2}+41 A \,b^{3} m^{2} x^{3}+B \,a^{3} m^{4} x +36 B \,a^{2} b \,m^{3} x^{2}+123 B a \,b^{2} m^{2} x^{3}+50 m \,x^{4} b^{3} B +A \,a^{3} m^{4}+39 A \,a^{2} b \,m^{3} x +147 A a \,b^{2} m^{2} x^{2}+61 A \,b^{3} x^{3} m +13 B \,a^{3} m^{3} x +147 B \,a^{2} b \,m^{2} x^{2}+183 B a \,b^{2} x^{3} m +24 b^{3} B \,x^{4}+14 A \,a^{3} m^{3}+177 A \,a^{2} b \,m^{2} x +234 a A \,b^{2} x^{2} m +30 A \,b^{3} x^{3}+59 B \,a^{3} m^{2} x +234 B \,a^{2} b \,x^{2} m +90 B a \,b^{2} x^{3}+71 A \,a^{3} m^{2}+321 a^{2} A b x m +120 a A \,b^{2} x^{2}+107 a^{3} B x m +120 B \,a^{2} b \,x^{2}+154 a^{3} A m +180 a^{2} A b x +60 a^{3} B x +120 a^{3} A \right )}{\left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(454\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 129, normalized size = 1.34 \begin {gather*} \frac {B b^{3} x^{m + 5}}{m + 5} + \frac {3 \, B a b^{2} x^{m + 4}}{m + 4} + \frac {A b^{3} x^{m + 4}}{m + 4} + \frac {3 \, B a^{2} b x^{m + 3}}{m + 3} + \frac {3 \, A a b^{2} x^{m + 3}}{m + 3} + \frac {B a^{3} x^{m + 2}}{m + 2} + \frac {3 \, A a^{2} b x^{m + 2}}{m + 2} + \frac {A a^{3} x^{m + 1}}{m + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 379 vs.
\(2 (96) = 192\).
time = 1.31, size = 379, normalized size = 3.95 \begin {gather*} \frac {{\left ({\left (B b^{3} m^{4} + 10 \, B b^{3} m^{3} + 35 \, B b^{3} m^{2} + 50 \, B b^{3} m + 24 \, B b^{3}\right )} x^{5} + {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} m^{4} + 90 \, B a b^{2} + 30 \, A b^{3} + 11 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m^{3} + 41 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m^{2} + 61 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m\right )} x^{4} + 3 \, {\left ({\left (B a^{2} b + A a b^{2}\right )} m^{4} + 40 \, B a^{2} b + 40 \, A a b^{2} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} m^{3} + 49 \, {\left (B a^{2} b + A a b^{2}\right )} m^{2} + 78 \, {\left (B a^{2} b + A a b^{2}\right )} m\right )} x^{3} + {\left ({\left (B a^{3} + 3 \, A a^{2} b\right )} m^{4} + 60 \, B a^{3} + 180 \, A a^{2} b + 13 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m^{3} + 59 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m^{2} + 107 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m\right )} x^{2} + {\left (A a^{3} m^{4} + 14 \, A a^{3} m^{3} + 71 \, A a^{3} m^{2} + 154 \, A a^{3} m + 120 \, A a^{3}\right )} x\right )} x^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2018 vs.
\(2 (87) = 174\).
time = 0.41, size = 2018, normalized size = 21.02 \begin {gather*} \begin {cases} - \frac {A a^{3}}{4 x^{4}} - \frac {A a^{2} b}{x^{3}} - \frac {3 A a b^{2}}{2 x^{2}} - \frac {A b^{3}}{x} - \frac {B a^{3}}{3 x^{3}} - \frac {3 B a^{2} b}{2 x^{2}} - \frac {3 B a b^{2}}{x} + B b^{3} \log {\left (x \right )} & \text {for}\: m = -5 \\- \frac {A a^{3}}{3 x^{3}} - \frac {3 A a^{2} b}{2 x^{2}} - \frac {3 A a b^{2}}{x} + A b^{3} \log {\left (x \right )} - \frac {B a^{3}}{2 x^{2}} - \frac {3 B a^{2} b}{x} + 3 B a b^{2} \log {\left (x \right )} + B b^{3} x & \text {for}\: m = -4 \\- \frac {A a^{3}}{2 x^{2}} - \frac {3 A a^{2} b}{x} + 3 A a b^{2} \log {\left (x \right )} + A b^{3} x - \frac {B a^{3}}{x} + 3 B a^{2} b \log {\left (x \right )} + 3 B a b^{2} x + \frac {B b^{3} x^{2}}{2} & \text {for}\: m = -3 \\- \frac {A a^{3}}{x} + 3 A a^{2} b \log {\left (x \right )} + 3 A a b^{2} x + \frac {A b^{3} x^{2}}{2} + B a^{3} \log {\left (x \right )} + 3 B a^{2} b x + \frac {3 B a b^{2} x^{2}}{2} + \frac {B b^{3} x^{3}}{3} & \text {for}\: m = -2 \\A a^{3} \log {\left (x \right )} + 3 A a^{2} b x + \frac {3 A a b^{2} x^{2}}{2} + \frac {A b^{3} x^{3}}{3} + B a^{3} x + \frac {3 B a^{2} b x^{2}}{2} + B a b^{2} x^{3} + \frac {B b^{3} x^{4}}{4} & \text {for}\: m = -1 \\\frac {A a^{3} m^{4} x x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {14 A a^{3} m^{3} x x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {71 A a^{3} m^{2} x x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {154 A a^{3} m x x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {120 A a^{3} x x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {3 A a^{2} b m^{4} x^{2} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {39 A a^{2} b m^{3} x^{2} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {177 A a^{2} b m^{2} x^{2} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {321 A a^{2} b m x^{2} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {180 A a^{2} b x^{2} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {3 A a b^{2} m^{4} x^{3} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {36 A a b^{2} m^{3} x^{3} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {147 A a b^{2} m^{2} x^{3} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {234 A a b^{2} m x^{3} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {120 A a b^{2} x^{3} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {A b^{3} m^{4} x^{4} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {11 A b^{3} m^{3} x^{4} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {41 A b^{3} m^{2} x^{4} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {61 A b^{3} m x^{4} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {30 A b^{3} x^{4} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {B a^{3} m^{4} x^{2} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {13 B a^{3} m^{3} x^{2} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {59 B a^{3} m^{2} x^{2} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {107 B a^{3} m x^{2} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {60 B a^{3} x^{2} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {3 B a^{2} b m^{4} x^{3} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {36 B a^{2} b m^{3} x^{3} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {147 B a^{2} b m^{2} x^{3} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {234 B a^{2} b m x^{3} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {120 B a^{2} b x^{3} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {3 B a b^{2} m^{4} x^{4} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {33 B a b^{2} m^{3} x^{4} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {123 B a b^{2} m^{2} x^{4} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {183 B a b^{2} m x^{4} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {90 B a b^{2} x^{4} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {B b^{3} m^{4} x^{5} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {10 B b^{3} m^{3} x^{5} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {35 B b^{3} m^{2} x^{5} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {50 B b^{3} m x^{5} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} + \frac {24 B b^{3} x^{5} x^{m}}{m^{5} + 15 m^{4} + 85 m^{3} + 225 m^{2} + 274 m + 120} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 593 vs.
\(2 (96) = 192\).
time = 2.14, size = 593, normalized size = 6.18 \begin {gather*} \frac {B b^{3} m^{4} x^{5} x^{m} + 3 \, B a b^{2} m^{4} x^{4} x^{m} + A b^{3} m^{4} x^{4} x^{m} + 10 \, B b^{3} m^{3} x^{5} x^{m} + 3 \, B a^{2} b m^{4} x^{3} x^{m} + 3 \, A a b^{2} m^{4} x^{3} x^{m} + 33 \, B a b^{2} m^{3} x^{4} x^{m} + 11 \, A b^{3} m^{3} x^{4} x^{m} + 35 \, B b^{3} m^{2} x^{5} x^{m} + B a^{3} m^{4} x^{2} x^{m} + 3 \, A a^{2} b m^{4} x^{2} x^{m} + 36 \, B a^{2} b m^{3} x^{3} x^{m} + 36 \, A a b^{2} m^{3} x^{3} x^{m} + 123 \, B a b^{2} m^{2} x^{4} x^{m} + 41 \, A b^{3} m^{2} x^{4} x^{m} + 50 \, B b^{3} m x^{5} x^{m} + A a^{3} m^{4} x x^{m} + 13 \, B a^{3} m^{3} x^{2} x^{m} + 39 \, A a^{2} b m^{3} x^{2} x^{m} + 147 \, B a^{2} b m^{2} x^{3} x^{m} + 147 \, A a b^{2} m^{2} x^{3} x^{m} + 183 \, B a b^{2} m x^{4} x^{m} + 61 \, A b^{3} m x^{4} x^{m} + 24 \, B b^{3} x^{5} x^{m} + 14 \, A a^{3} m^{3} x x^{m} + 59 \, B a^{3} m^{2} x^{2} x^{m} + 177 \, A a^{2} b m^{2} x^{2} x^{m} + 234 \, B a^{2} b m x^{3} x^{m} + 234 \, A a b^{2} m x^{3} x^{m} + 90 \, B a b^{2} x^{4} x^{m} + 30 \, A b^{3} x^{4} x^{m} + 71 \, A a^{3} m^{2} x x^{m} + 107 \, B a^{3} m x^{2} x^{m} + 321 \, A a^{2} b m x^{2} x^{m} + 120 \, B a^{2} b x^{3} x^{m} + 120 \, A a b^{2} x^{3} x^{m} + 154 \, A a^{3} m x x^{m} + 60 \, B a^{3} x^{2} x^{m} + 180 \, A a^{2} b x^{2} x^{m} + 120 \, A a^{3} x x^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.60, size = 289, normalized size = 3.01 \begin {gather*} \frac {A\,a^3\,x\,x^m\,\left (m^4+14\,m^3+71\,m^2+154\,m+120\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {B\,b^3\,x^m\,x^5\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {a^2\,x^m\,x^2\,\left (3\,A\,b+B\,a\right )\,\left (m^4+13\,m^3+59\,m^2+107\,m+60\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {b^2\,x^m\,x^4\,\left (A\,b+3\,B\,a\right )\,\left (m^4+11\,m^3+41\,m^2+61\,m+30\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {3\,a\,b\,x^m\,x^3\,\left (A\,b+B\,a\right )\,\left (m^4+12\,m^3+49\,m^2+78\,m+40\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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